On the interpolation with the potential bound for global solutions of the defocusing cubic wave equation on T2

We prove that the solutions of the defocusing cubic wave equation on T2 exist globally in time in Hs(T2) for s > 2/5 by contradiction. Assuming that one of the maximal times of existence is finite, we prove that the Sobolev norm of each of these solutions is bounded in an open neighborhood of it by estimating the growth of a mollified energy through the interpolation with the potential bound for the low frequency part.